Exercise 5.1.40

For a category and object , let denote the full subcategory of whose objects are the monic morphisms. A subobject of is an isomorphism class of objects in .

Theorem: Let and be monics in .

So, we have that subobjects of a set are in canonical correspondence with the subsets of .

In , the situation calls for a bit more subtlety. There exist monics in that that have the same subspace as an image, but are not isomorphic. Consider the space and . In both cases, consider the map . So we have and that are monics with the same image (), but and are not homeomorphic, and hence these are two different subobjects. However, isomorphism classes of embeddings (a continuous injective function that is homeomorphic onto its image) do correspond one-to-one with subspaces.